(03/22/2010, 10:14 AM)mike3 Wrote: (BTW, I've been playing around with another tetration method based on trying to use the Borel summation on Ansus' continuum-sum formula. If you want, I can post some rough observations from an attempt at numerical approximation. I'm still not sure if it converges, as it seems to take tons of precision and terms to work, so I can't really press past more than a few decimals of accuracy.)

Please do that. (At least) I would be very interested to read more about this.

(03/08/2010, 11:59 AM)bo198214 Wrote: ...So far it seems these pertubed Fatou coordinates only give real analytic functions for conjugated complex fixed point pairs. To give it a distinguished name (and as "perturbured Fatou coordinates" does not really hit the point) I will call it from here bipolar iteration in mnemonic that the Abel function is defined on a sickel between *two* fixed points, in opposite to the regular iteration which is defined in a neighborhood of *one* fixed point.
.....
Second, the bipolar Fatou coordinates may exist between two real fixed points (though even that is not completely clear to me and not backed by the theory) but it may not be real analytic.

So the question is whether there is at all a real tetration on (1,oo) which is particularly analytic in the base at e^(1/e) ...

Any updates on whether the theory supports bipolar Fatou solutions between two real fixed points? This would be analogous to the tetration solution, in my complex base solution post, where I propose the bipolar solution continues along a circle past the Shell Thron boundary, where one of the fixed points switches from repelling to attracting.

(01/25/2010, 07:30 PM)bo198214 Wrote: ...start with a parabolic fixed point of a holomorphic function , i.e. and .
If we slightly perturb this function by a complex , , then the one fixed point splits up into several fixed points (at least 2). But the perturbed function still behaves quite similar to the original function.

It would seem that the bipolar Fatou solutions should have encountered this exact question. I very briefly experimented with some of these ideas applied to a simple function like , which has a parabolic fixed point at z=0, for epsilon=0, and it should have a real valued superfunction if is a postive real number, which behaves similar to tetration for real bases in many ways.

(01/25/2010, 07:30 PM)bo198214 Wrote: ...If we slightly perturb this function by adding ... then the one parabolic fixed point splits up into two conjugate repelling fixed points. If we slightly perturb with then the parabolic fixed point splits up into two real fixed points on the real axis (one attracting, one repelling).

So, just as in the tetration solution, the perturbed Fatou method generates a bipolar superfunction if both fixed points are repelling. And, like tetration, if epsilon starts out as a positive real number, then as epsilon goes around in a circle around 0 counterclockwise, very soon one of the two repelling fixed points switches to attracting. Also, of course, there is the case where epsilon is exactly on the transition, with one repelling fixed point, and the other fixed point rationally or irrationally indifferent, which would be analogous to the Shell Thron boundary. As is varied, has such a boundary, which seems very similar to the Shell Thron boundary for tetration.

It would be interesting to know what the theory has to say about bipolar solutions on that boundary, and bipolar solutions inside the boundary. Also, mathematicians interested in perturbed Fatou coordinates, and parabolic implosion should have a natural interest in tetration, as an example of a bipolar Fatou solution.
- Sheldon

I found a good paper online by Mitsushiro Shishikura and Hiroyuki Inou titled, The renormalization for parabolic fixed points and their perturbation. I'm reading both this paper, as well as Shishikura's chapter "Bifurcation of Parabolic Fixed Points", in the book "The Mandelbrot Set, Theme and Variations". The online paper is more recent, and probably improved. I haven't gotten very far yet, but I see some concepts that seem similar to the 1-cyclic theta transformations I have used in my posts on this forum, which seems to correspond to Shishikura's horn map. Then the Tetration programs I have posted here which are iteratively calculating and sexp(z), are hopefully calculating what is mathematically equivalent to Shishikura's horn map, for . The notation used by Shishikura for the horn map is:, where is periodic with period 1, and decays to a constant as .
- Sheldon

6. (Douady) Persistence of the Fatou Coordinate:
First consider the polynomial for a real small . This polynomial has two repelling fixed points z1,z2 such that . Let be the vertical line connecting them. is a curve connecting the fixed points to the right of .

We define a Fatou coordinate on a neighborhood that contains , the region between them, but NOT the endpoints z1; z2 as the map satisfying.

We can vary into the complex plane. As its argument increases we can follow . The deformed curves and spiral into the fixed points. As 1/4 + crosses into the main cardioid of the Mandelbrot set, one of the fixed points, say z1, becomes attracting. is still defined. The curves spiral in at the attracting fixed point; is attracted "inside" . When we reach the boundary of the cardioid again (with argument of ), the spiral breaks. That is, because the fixed point becomes indifferent, the spiral is no longer defined; cannot be attracted "inside" . Thus the continuation of the Fatou coordinate breaks down at the second crossing (below the real axis) of the cardioid.

The construction can be made exactly the same way by making the argument of negative. Thus there are two overlapping regions of the parameter plane in which a Fatou coordinate is defined. Each can only be defined for a single crossing of the cardioid.

The Fatou coordinate can be defined near a parabolic point for an arbitrary rational or entire map. The question is to understand the obstruction to extending the coordinate in this setting. Is the obstruction local or global?

I don't know the status of Douady's problem. In his question, is the Abel function or Fatou coordinate near a parabolic fixed point, analogous to the Fatou coordinate, for the bipolar tetration solution for where . One can compare Douady's problem above, with my post a few weeks ago, on the analogous problem for tetration. Both problems behave similarly, and both may have analogous Riemann surfaces. Also, I have generated some solutions for the superfunction, for , and there is some pretty bizarre behavior if corresponds to an indifferent fixed point with a rational period, corresponding somewhat to pictures I posted in this post, for the analogous tetration bipolar solution where the fixed point has a period=5.

If anyone else has any related information, or publications, I'd be very interested.
- Sheldon Levenstein

(03/07/2012, 12:08 AM)sheldonison Wrote: ....
But first, I briefly want to describe the merged tetration solution and the Shell Thron boundary, and briefly describe reasons why one might not expect there to be solutions at the Shell Thron boundary boundary. Briefly, start with real tetration at the real axis, for bases>eta=e^(1/e), and rotate either clockwise, or counter clockwise. Initially, both fixed points are repelling. Consider the case where we rotate counterclockwise, through increasing imag(z). Initially, the period for the upper fixed point will have positive real and imaginary components. At the Shell Thron boundary, the upper fixed point has a real period; the lower fixed point is still repelling. I call this the first Shell Thron boundary crossing. Before continuing, I would briefly state that as we continue rotating around eta counterclockwise, we will reach the real axis for points<eta, and then continuing further, we will reach the Shell Thron boundary a second time. I believe there is a singularity at the second Shell Thron boundary crossing, but there is not a singularity at the first Shell Thron boundary crossing.
....

6. (Douady) Persistence of the Fatou Coordinate:
First consider the polynomial for a real small . This polynomial has two repelling fixed points z1,z2 such that . Let be the vertical line connecting them. is a curve connecting the fixed points to the right of .

We define a Fatou coordinate on a neighborhood that contains , the region between them, but NOT the endpoints z1; z2 as the map satisfying.

We can vary into the complex plane. As its argument increases we can follow . The deformed curves and spiral into the fixed points. As 1/4 + crosses into the main cardioid of the Mandelbrot set, one of the fixed points, say z1, becomes attracting. is still defined. The curves spiral in at the attracting fixed point; is attracted "inside" . When we reach the boundary of the cardioid again (with argument of ), the spiral breaks. That is, because the fixed point becomes indifferent, the spiral is no longer defined; cannot be attracted "inside" . Thus the continuation of the Fatou coordinate breaks down at the second crossing (below the real axis) of the cardioid.

The construction can be made exactly the same way by making the argument of negative. Thus there are two overlapping regions of the parameter plane in which a Fatou coordinate is defined. Each can only be defined for a single crossing of the cardioid.

The Fatou coordinate can be defined near a parabolic point for an arbitrary rational or entire map. The question is to understand the obstruction to extending the coordinate in this setting. Is the obstruction local or global?

I don't know the status of Douady's problem. In his question, is the Abel function or Fatou coordinate near a parabolic fixed point, analogous to the Fatou coordinate, for the bipolar tetration solution for where . One can compare Douady's problem above, with my post a few weeks ago, on the analogous problem for tetration. Both problems behave similarly, and both may have analogous Riemann surfaces. Also, I have generated some solutions for the superfunction, for , and there is some pretty bizarre behavior if corresponds to an indifferent fixed point with a rational period, corresponding somewhat to pictures I posted in this post, for the analogous tetration bipolar solution where the fixed point has a period=5.

If anyone else has any related information, or publications, I'd be very interested.
- Sheldon Levenstein

(03/07/2012, 12:08 AM)sheldonison Wrote: ....
But first, I briefly want to describe the merged tetration solution and the Shell Thron boundary, and briefly describe reasons why one might not expect there to be solutions at the Shell Thron boundary boundary. Briefly, start with real tetration at the real axis, for bases>eta=e^(1/e), and rotate either clockwise, or counter clockwise. Initially, both fixed points are repelling. Consider the case where we rotate counterclockwise, through increasing imag(z). Initially, the period for the upper fixed point will have positive real and imaginary components. At the Shell Thron boundary, the upper fixed point has a real period; the lower fixed point is still repelling. I call this the first Shell Thron boundary crossing. Before continuing, I would briefly state that as we continue rotating around eta counterclockwise, we will reach the real axis for points<eta, and then continuing further, we will reach the Shell Thron boundary a second time. I believe there is a singularity at the second Shell Thron boundary crossing, but there is not a singularity at the first Shell Thron boundary crossing.
....

that as the STR boundary is approached for the second crossing, the lines of singularities in the upper half plane get denser. It seems that when we hit the boundary, they should become "infinitely dense", killing the holomorphizm and apparently signalling the reaching of a natural boundary of analyticity at the STR boundary. This behavior would be much akin to that of the single-fixpoint regular iterations, though in this case it does not prevent the existence of a branch (e.g. the principal one) that is defined over the whole plane minus some singularities (and a cut, if it must be continuous over its whole domain).

Hmm, now that's odd. I tried walking a base around and monitoring the sickel to see what happened to it. I walked it from up to , left to , down to and finally right to , crossing the STR boundary twice. I noticed nothing weird at the second boundary crossing...

EDIT: I've continued it further, to and then back up to , and noticed nothing "singularitylike". However, I did find that on returning to , I was once again at a sickel between the two principal fixed points, only now they had been swapped. Wtf? Does this mean there's a merged tetrational using the two fixed points in opposite order? That'd be very bizarre. Or, perhaps, this in fact indicates that there is a failure somewhere (probably at the STR boundary) and the continuation is in fact not possible.

(04/17/2012, 07:51 AM)mike3 Wrote: ....
EDIT: I've continued it further, to and then back up to , and noticed nothing "singularitylike". However, I did find that on returning to , I was once again at a sickel between the two principal fixed points, only now they had been swapped. Wtf? Does this mean there's a merged tetrational using the two fixed points in opposite order? That'd be very bizarre. Or, perhaps, this in fact indicates that there is a failure somewhere (probably at the STR boundary) and the continuation is in fact not possible.

edited Where both fixed points are repelling, there is only one solution, and the fixed points cannot be swapped (check your math, and/or else post of sexp(z)). Henryk has a paper with a uniquess criteria for sexp(z), and I hope to eventually show the theta(z) uniqueness criteria I posted yesterday,, can be shown to be complete if both fixed points are repelling and theta(z) has a singularity.

But, inside the Shell Thron boundary, where one of the fixed points is attracting, there are two different solutions for the same base, which also makes the uniqueness criteria more complicated. Are you calculating sexp(z), or slog(z)? I've started to make plots of slog(z) as well, for the bipolar solutions, inside the Shell Thron boundary. Also, as you point out, as you approach the second Shell Thron crossing, the singularities for sexp(z) in the upper half of the complex plane (rotating around eta counterclockwise) start to bunch up and get arbitrarily close to each other and the real axis.
- Sheldon